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Undergraduate chapters and video availability01 · Read and understand
What you will learn
- Compute a likelihood ratio while keeping hypotheses distinct from posterior probabilities.
- Justify the conclusion "The ratio is 27/16" using the stated assumptions.
Before you start
Bernoulli trials and likelihood.
Keep paper nearby. Read the question once for the context, then again to identify what is known and what you need to find.
Start with a question
In four independent Bernoulli trials with three successes, compare p=3/4 with p=1/2.
Why this math matters
Compute a likelihood ratio while keeping hypotheses distinct from posterior probabilities. This worked micro-lesson connects a precise mathematical condition to a conclusion you can check. The transfer task asks you to change the setting and decide which parts of the reasoning still apply.
Set up the model
A useful answer starts with clear assumptions:
- The trials are independent with one common p.
- The observed count is the same under both candidate models.
02 · Work through the example
Follow the reasoning, one step at a time.
Try to predict the next step before reading it. After each calculation, explain why the operation makes sense and how it helps answer the original question.
Work through it with Amar
See this example unfold.
The complete worked example, one idea at a time.
Compare two parameter values using observed evidence
PausedQuestion: Start with the question. Paused.
Question
Start with the question
In four independent Bernoulli trials with three successes, compare p=3/4 with p=1/2.
Before you calculate
Read what is known and what you need to find. Make a prediction before moving to the first calculation.
Starts paused. Play advances through the full text at a reading pace; pause whenever you need more time. Previous, Next, and the phase buttons let you set your own pace. Playback pauses when this walkthrough leaves the screen or you switch tabs.
Your device’s reduced-motion setting keeps each phase still. Manual controls remain available. The full written solution stays below.
Build the model
For the observed success count, L(p) is proportional to p³(1−p)
The common binomial coefficient cancels in a ratio.
Work through the mathematics
L(3/4)/L(1/2)=[(3/4)³(1/4)]/(1/16)
Substitute the two candidate values into the same likelihood.
Check the conclusion
The ratio is 27/16
These data are more likely under p=3/4 by this factor, but this is not itself posterior odds without prior odds.
The result
The ratio is 27/16
These data are more likely under p=3/4 by this factor, but this is not itself posterior odds without prior odds.
Common mistakes to catch
- Do not include different combinatorial factors in numerator and denominator.
- Evidence and prior belief play separate roles.
03 · Practice independently
Try it before revealing the answer.
Use paper or a calculator as needed. Write your units and reasoning, then open the hint or explanation to check your approach.
Practice 1
What happens if all four trials are failures?
Show a hint
Use likelihood (1−p)⁴.
Reveal answer and explanation
The ratio becomes 1/16
(1/4)⁴/(1/2)⁴=1/16 favors p=1/2.
Practice 2
Does a likelihood ratio of two mean probability two for a hypothesis?
Show a hint
Ratios are not normalized probabilities.
Reveal answer and explanation
No
It compares data probabilities under two models.
Take the idea with you
Report a likelihood comparison without turning it into an unsupported certainty claim.
04 · Reflect and continue
Can you explain it in your own words?
Before moving on, explain the main idea without looking at the worked example. Try both practice questions, check your reasoning, and name one mistake you now know how to avoid. Return to a step if you still need support.
Up next: Quantify local information about a Bernoulli probability
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